section10mm
0.70.5Preliminaries
We first list a basic inequality concerning the time-space integrable estimation with the square of the heat kernel as a weight function,
which will play an important role in the sequel.
This result is borrowed from [1]} and we skip the details here.
For \( \alpha > 0 \) , we define
\(\omega (x,t)=\left( 1+t \right)^{-\frac{1}{2}}\mathrm {e}^{-\frac{\alpha x^2}{1+t}},\qquad g(x,t) = \int _{-\infty }^x \omega (y,t)\,\mathrm {d}y,\) 
